2x squard - 5x cubed +7x to the 4th power - 9

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Supposing that you need to solve for x the equation `2x^2 - 5x^3 + 7x^4 - 9 = 0` , you need to arrange the terms such that:

`7x^4 - 5x^3 + 2x^2 - 9 = 0`

You need te check if there exists a root between the fractions `+-3/7;+-9/7` such that:

`7*(3/7)^4 - 5*(3/7)^3 + 2*(3/7)^2 - 9 = 0`

`(3/7)^2(7*(3/7)^2 - 5*(3/7) + 2) - 9 = 0`

`9/49(63/49 - 15/7 + 2) - 9 = 0 `

`9/49(-42/49 + 2) - 9 = 0 => 9*56/49*49 - 9!= 0`

Notice that `x = +3/7` are not the roots of the given equation.

Sketching the graph of the polynomial yields that the curve intercepts x axis at a value of `x in (-1,-0.5)` and `x in (1,1.5),` hence, the equation has, either two roots of order of second order of multiplicity, or two real toots and two complex conjugate roots.